Rational points of lattice ideals on a toric variety and toric codes

We show that the number of rational points a subgroup inside toric variety over finite field defined by homogeneous lattice ideal can be computed via Smith normal form matrix whose columns constitute basis lattice. This generalizes and yields concise geometric proof same fact proven purely algebraically Lopez Villarreal for case projective space standard dimension one. also prove Nullstellensatz type theorem establishing one to correspondence between subgroups dense split torus certain ideals. As application, we compute main parameters generalized codes on Hirzebruch surfaces, generalizing existing literature.